I'll betcha if she catches the Doctor snuggling up to another TARDIS, she'll see Red

And, she doesn't have a defective Chameleon Circuit, she's just partial to the Blue Police Box look. (She did have a defective Chameleon Circuit when 6 started messing around with it, trying to force her to change, though).

There was also a red one in east London - it was used as the fire control station for one of the Thames tunnels (the Blackwall tunnel ISTR... not there now, was until the mid 1980s), and as it was run by the fire brigade it was painted red.

When I was a kid, a poster magazine used a red police box as a graphic: cue massive playground arguments about what colour the TARDIS really was, ending in every kid in the street cramming into the one house with a colour TV to watch Planet of the Spiders (I think) and see for themselves.

A scene where the Doctor is fiddling with the circuit, bangs it with his sonic screwdriver in frustration and the Tardis briefly turns red would be great...

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I am reminded of the scene in one of the Simpson's Halloween Specials where their house keeps changing because of the "damage" that Homer is doing to the timeline by the stuff he is doing in the past, thanks to the Toaster/Time-Machine.

Maybe the Doctor would be totally oblivious, but the audience would see it on-screen. It would only last for a few moments. Working under the console on some system supposedly unrelated to the Chameleon Circuit as the ship floats in deep space, he does not see how the TARDIS flashes from one outer shell to the next to the next, perhaps the first outer shell would simply be a red police box.... eventually back to the blue police box.

But your idea of a more brief, more simple segment might create a sweeter little moment on screen!

It reminds me of the alternate ending I'd dreamed up for "Doomsday" way back when: After the Doctor says goodbye to Rose on the beach, she suddenly hears a familiar wooshing sound. She turns around and sees a red police box. Christopher Eccleston walks out, dressed in a Tom Baker-ish wardrobe, looking with a bewildered expression at a large fold-out map. He introduces himself to Rose as "the Professor" and asks her if she knows how to get to the Eye of Orion.